System, method and apparatus for controlling converters using input-output linearization

ABSTRACT

A system, method and apparatus for controlling boost and buck-boost converters using input-output linearization and leading-edge modulation is provided. The controller includes a summing circuit connected to the converter to create a third voltage representing a difference between the first voltage and the second voltage. A gain circuit is connected to the summing circuit to adjust the third voltage by an appropriate gain. A modulating circuit is connected to the gain circuit, the converter, the first voltage, the second voltage and the second current to create a control signal based on the first voltage, the second voltage, the adjusted third voltage, the fourth voltage and the first current. The control signal is used to control the converter. Typically, the first voltage is a converter output voltage, the second voltage is a reference voltage, the fourth voltage is a converter input voltage, and first current is a converter inductor current.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of application Ser. No. 12/487,242,filed Jun. 18, 2009. The patent application identified above isincorporated herein by reference in its entirety to provide continuityof disclosure.

FIELD OF THE INVENTION

The present invention relates generally to providing modulation signalsto electrical circuits and, more particularly, to a system, method andapparatus for controlling converters using input-output linearizationand leading-edge modulation.

BACKGROUND OF THE INVENTION

Power converters are used to convert one form of energy to another(e.g., AC to AC, AC to DC, DC to AC, and DC to DC) thereby making itusable to the end equipment, such as computers, automobiles,electronics, telecommunications, space systems and satellites, andmotors. Every application of power electronics involves some aspect ofcontrol. Converters are typically identified by their capability and/orconfigurations, such as, buck converters, boost converters, buck-boostconverters, boost-buck converters (Ćuk), etc. For example, DC-DCconverters belong to a family of converters known as “switchingconverters” or “switching regulators.” This family of converters is themost efficient because the conversion elements switch from one state toanother, rather than needlessly dissipating power during the conversionprocess. Essentially there is a circuit with switches and twoconfigurations (each can be modeled as linear systems) in which theconverter resides according to the switch positions. The duty ratio (d)is the ratio indicating the time in which a chosen switch is in the “on”position while the other switch is in the “off” position, and this d isconsidered to be the control input. Input d is usually driven bypulse-width-modulation (PWM) techniques.

Switching from one state to another and the accompanying nonlinearity ofthe system causes problems. State space averaging reduces the switchingproblems to make the system, in general, a nonlinear averaged system fora boost converter or a buck-boost converter. But, control of the systemunder these nonlinear effects becomes difficult when certain performanceobjectives must be met. For the most part linearization is done througha Taylor series expansion. Nonlinear terms of higher orders are thrownaway and a linear approximation replaces the nonlinear system. Thislinearization method has proven effective for stabilizing control loopsat a specific operating point. However, use of this method requiresmaking several assumptions, one of them being so-called “small signaloperation.” This works well for asymptotic stability in the neighborhoodof the operating point, but ignores large signal effects which canresult in nonlinear operation of the control loop when, for example, anamplifier saturates during startup, or during transient modes, such asload or input voltage changes. Once nonlinear operation sets in, thecontrol loop can have equilibrium points unaccounted for in thelinearization.

One of the most widely used methods of pulse-width modulation istrailing-edge modulation (TEM), wherein the on-time pulse begins on theclock and terminates in accordance with a control law. Unstable zerodynamics associated with TEM in the continuous conduction mode (CCM)prevent the use of an input-output feedback linearization because itwould result in an unstable operating point. The other control method isleading-edge modulation (LEM), wherein the on-time pulse begins inaccordance with a control law and terminates on the clock. Thedifference between LEM and TEM is that in TEM the pulse-width isdetermined by the instantaneous control voltage v_(c) prior to switchturn-off, whereas in LEM the pulse-width is determined by v_(c), priorto switch turn-on.

There is, therefore, a need for a system, method and apparatus forcontrolling converters using input-output linearization that does notconstrain stability to one operating point, but rather to a set ofoperating points spanning the expected range of operation during startupand transient modes of operation

SUMMARY OF INVENTION

The present invention provides a system, method and apparatus forcontrolling converters using input-output linearization that does notconstrain stability to one operating point, but rather to a set ofoperating points spanning the expected range of operation during startupand transient modes of operation. In particular, the present inventionuses leading edge modulation and input output linearization to computethe duty ratio of a boost converter or a buck-boost converter. Thepresent invention can also be applied to other converter types.Moreover, the parameters in this control system are programmable, andhence the algorithm can be easily implemented on a DSP or in silicon,such as an ASIC.

Notably, the present invention provides at least four advantagescompared to the dominant techniques currently in use for powerconverters. The combination of leading-edge modulation and input-outputlinearization provides a linear system instead of a nonlinear system. Inaddition, the “zero dynamics” becomes stable because the zeros of thelinear part of the system are in the open left half plane. The presentinvention is also independent of stabilizing gain, as well as desiredoutput voltage or desired output trajectory.

More specifically, the present invention provides a system that includesa boost or buck-boost converter having a first voltage at an output ofthe converter and a first current at an inductor within the converter, areference voltage source having a second voltage, a fourth voltage froma voltage source providing an input voltage to the converter, and a PWMmodulator/controller. The PWM modulator/controller includes a summingcircuit connected to the converter and the reference voltage source tocreate a third voltage representing a difference between the firstvoltage from the output of the converter and the second voltage from thereference voltage source. A gain circuit is connected to the summingcircuit to adjust the third voltage by a proportional gain or by anysuitable type of controller, such as proportional (P), integral (I) orderivative (D) (or any combination of these three) controller. Amodulating circuit is connected to the gain circuit, the converter tocreate a control signal that provides leading-edge modulation withinput-output linearization based on the first voltage, the secondvoltage from the reference voltage source, the adjusted third voltagefrom the gain circuit, the fourth voltage from the voltage source or theinput of the converter, and the first current from the inductor withinthe converter. The control signal is then used to control the converter.Whenever the converter is a boost converter, the control signal has aduty cycle defined by

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$Whenever the converter is a buck-boost converter, the control signal hasa duty cycle defined by

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$Note that the second voltage source can be integrated into the PWMmodulator/controller. In addition, the PWM modulator/controller can beimplemented using a digital signal processor, a field programmable gatearray (FPGA) or conventional electrical circuitry. Moreover, theconverter can be controlled with a proportional controller, or anysuitable type of controller, such as a proportional (P), integral (I) orderivative (D) (or any combination of these three) controller, byreplacing k(y₀−y) in the equation defining the duty cycle d with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller.

The present invention also provides an apparatus that includes one ormore electrical circuits that provide a control signal to a boostconverter such that a duty cycle of the control signal is defined as

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$Similarly, the present invention provides an apparatus that includes oneor more electrical circuits that provide a control signal to abuck-boost converter such that a duty cycle of the control signal isdefined as

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$In either case, the apparatus may include a summing circuit, a gaincircuit, a modulating circuit and various connections. The connectionsinclude a first connection to receive a first voltage from an output ofthe converter, a second connection to receive a second voltage from areference voltage source, a third connection to receive a first currentfrom an inductor within the converter, a fourth connection to receive aninput voltage from a voltage source providing an input to converter anda fifth connection to output a control signal to the converter. Thesumming circuit is connected to the first connection and the secondconnection to create a third voltage representing a difference betweenthe first voltage from the output of the converter and the secondvoltage from the reference voltage source. The gain circuit is connectedto the summing circuit to adjust the third voltage by a proportionalgain or by any suitable type of controller, such as proportional (P),integral (I) or derivative (D) (or any combination of these three)controller. The modulating circuit is connected to the gain circuit, thesecond connection, the third connection, the fourth connection and thefifth connection. The modulation circuit creates a control signal thatprovides leading-edge modulation with input-output linearization basedon the first voltage from the output of the converter, the secondvoltage from the reference voltage source, the adjusted third voltagefrom the gain circuit, the fourth voltage from the voltage source or theinput of the converter, and the first current from the inductor withinthe converter.

Moreover, the present invention can be sold as a kit for engineers todesign and implement a PWM modulated converter (boost or buck-boost).The kit may include a digital signal processor, or field programmablegate array (FPGA), and a computer program embodied on a computerreadable medium for programming the digital signal processor, or FPGA,to control the PWM modulated converter. The computer program may alsoinclude one or more design tools. The digital signal processor, or FPGA,includes a summing circuit, a gain circuit, a modulating circuit andvarious connections. The various connections include a first connectionto receive a first voltage from an output of the converter, a secondconnection to receive a second voltage from a reference voltage source,a third connection to receive a first current from an inductor withinthe converter, a fourth connection to receive the input voltage from avoltage source providing an input to converter, and a fifth connectionto output a control signal to the converter. The summing circuit isconnected to the first connection and the second connection to create athird voltage representing a difference between the first voltage fromthe output of the converter and the second voltage from the referencevoltage source. The gain circuit is connected to the summing circuit toadjust the third voltage by a proportional gain or by any suitable typeof controller, such as proportional (P), integral (I) or derivative (D)(or any combination of these three) controller. The modulating circuitis connected to the gain circuit, the second connection, the thirdconnection, the fourth connection and the fifth connection. Themodulation circuit creates a control signal that provides leading-edgemodulation with input-output linearization based on the first voltagefrom the output of the converter, the second voltage from the referencevoltage source, the adjusted third voltage from the gain circuit, fourthvoltage from the voltage source or the input of the converter, and afirst current from the inductor within the converter. Whenever theconverter is a boost converter, the control signal has a duty cycledefined by

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$Whenever the converter is a buck-boost converter, the control signal hasa duty cycle defined by

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$

Furthermore, the present invention provides a method for controlling aboost or buck-boost converter using a PWM modulator/controller byreceiving a first voltage from an output of the converter, a secondvoltage from a reference voltage source, a first current from aninductor within the converter, and creating a third voltage representinga difference between the first voltage from the output of the converterand the second voltage from the reference voltage source and a fourthvoltage from a voltage source or the input of the converter. The thirdvoltage is then adjusted by a proportional gain or by any suitable typeof controller, such as proportional (P), integral (I) or derivative (D)(or any combination of these three) controller. The control signal iscreated that provides leading-edge modulation with input-outputlinearization based on the first voltage from the output of theconverter, the second voltage from the reference voltage source, theadjusted third voltage, the fourth voltage from the voltage source orthe input of the converter, and the first current from the inductorwithin the converter. The converter is then controlled using the controlsignal created by the PWM modulator/controller. Whenever the converteris a boost converter, the control signal has a duty cycle defined by

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$Whenever the converter is a buck-boost converter, the control signal hasa duty cycle defined by

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$

Note that the converter can be controlled with a proportionalcontroller, or any suitable type of controller, such as proportional(P), integral (I) or derivative (D) (or any combination of these three)controller, by replacing k(y₀−y) in the equation defining the duty cycled with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller. Moreover, the controlsignal can be created using a first order system, or can be independentof a stabilizing gain, a desired output voltage or a desired outputtrajectory. Likewise, the present invention may include a computerprogram embodied within a digital signal processor, or FPGA, wherein thesteps are implemented as one or more code segments.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and further advantages of the invention may be betterunderstood by referring to the following description in conjunction withthe accompanying drawings.

FIG. 1 is a block diagram of a system in accordance with the presentinvention.

FIG. 2 is a block diagram of a modulator/controller in accordance withthe present invention.

FIG. 3A is a flow chart of a method for controlling a boost converterusing a PWM modulator/controller in accordance with the presentinvention.

FIG. 3B is a flow chart of a method for controlling a buck-boostconverter using a PWM modulator/controller in accordance with thepresent invention.

FIGS. 4A and 4B are graphs of trailing-edge modulation of a PWM signaland leading-edge modulation of a PWM

FIG. 5 is circuit diagram of a boost converter and amodulator/controller in accordance with the present invention.

FIGS. 6A and 6B are linear circuit diagrams of a boost converter duringtime DTs and D′Ts, respectively in accordance with the presentinvention.

FIG. 7 is a graph of typical waveforms for the boost converter for thetwo switched intervals DTs and D′Ts in accordance with the presentinvention.

FIG. 8 is circuit diagram of a buck-boost converter and amodulator/controller in accordance with the present invention.

FIGS. 9A and 9B are linear circuit diagrams of a buck-boost converterduring time DTs and D′Ts, respectively in accordance with the presentinvention.

FIG. 10 is a graph of typical waveforms for the buck-boost converter forthe two switched intervals DTs and D′Ts in accordance with the presentinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

In the description that follows, like parts are marked throughout thespecification and figures with the same numerals, respectively. Thefigures are not necessarily drawn to scale and may be shown inexaggerated or generalized form in the interest of clarity andconciseness.

While the making and using of various embodiments of the presentinvention are discussed in detail below, it should be appreciated thatthe present invention provides many applicable inventive concepts thatcan be embodied in a wide variety of specific contexts. The specificembodiments discussed herein are merely illustrative of specific ways tomake and use the invention and do not delimit the scope of theinvention.

The present invention provides a system, method and apparatus forcontrolling converters using input-output linearization that does notconstrain stability to one operating point, but rather to a set ofoperating points spanning the expected range of operation during startupand transient modes of operation. In particular, the present inventionuses leading-edge modulation and input-output linearization to computethe duty ratio of a boost converter or a buck-boost converter. Thepresent invention can also be applied to other converter types.Moreover, the parameters in this control system are programmable, andhence the algorithm can be easily implemented on a DSP or in silicon,such as an ASIC.

Notably, the present invention provides at least four advantagescompared to the dominant techniques currently in use for powerconverters. The combination of leading-edge modulation and input-outputlinearization provides a linear system instead of a nonlinear system. Inaddition, the “zero dynamics” becomes stable because the zeros of thelinear part of the system are in the open left half plane. The presentinvention is also independent of stabilizing gain, as well as desiredoutput voltage or desired output trajectory.

As previously described, trailing-edge modulation for boost andbuck-boost converters operating in the continuous conduction mode givesrise to unstable zero dynamics where the linear part of the system aboutan operating point has a right half plane zero. In contrast, the presentinvention employs leading-edge modulation, along with some very simpledesign constraints, that change the zero dynamics so that the linearpart of the system has only open left half plane zeros. Since thenonlinear system now possesses stable zero dynamics, input-outputfeedback linearization can be used. To apply this method, the actualoutput y is chosen as output function h(x), and y is repeatedlydifferentiated until the input u appears. The number ofdifferentiations, r, is called the relative degree of the system. Thepresent invention has a relative degree r=1. The linearizingtransformation for d is solved and used for the control input. Thistransformation is local in nature, but it can be applied in aneighborhood of any state space operating point in DC-DC conversion.

It is desirable to choose any operating point for the nonlinear system.This operating point can be made locally asymptotically stable by theabove process if a gain k is chosen to be positive. The gain k does nothave to be adjusted for each operating point, i.e., no gain schedulingis required. However, the reference input will have to be walked up,which is typical of soil-start operation, to insure convergence to theoperating point. Note that Proportional (P), Integral (I), Derivative(D), Proportional-Integral (PI), and Proportional-Integral-Derivative(PID) control loops can be added for robustness.

Now referring to FIG. 1, a block diagram of a system 100 in accordancewith the present invention is shown. The system includes a power source(voltage) 102 connected to a converter 104 that provides power to a load106. The converter 104 is either a boost converter or a buck-boostconverter. The converter 104 is also connected to the PWMmodulator/controller 108. The PWM modulator/controller 108 receives afirst voltage 110 from an output of the converter 104, a second voltage(reference voltage) 112 from a voltage reference source (not shown), afirst current 114 from an inductor within the converter, and a fourthvoltage 116 from the voltage source 102 (i.e., the input voltage to theconverter 104). A summing circuit within the PWM modulator andcontroller 108 creates a third voltage representing the differencebetween the first voltage 110 from the output of the converter 104 andthe second voltage 112 from the reference voltage source. Note that thesource of the second voltage 112 (voltage reference source) can beintegrated within or external to the PWM modulator/controller 108. ThePWM modulator/controller 108 uses the first voltage 110, the secondvoltage 112, the first current 114 and the fourth voltage 116 togenerate a control signal 118 that is used to control the converter 104.The details of how the PWM modulator/controller 108 generates thecontrol signal 118 will be described in more detail below. In addition,the PWM modulator/controller 108 can be implemented using a digitalsignal processor, an FPGA, or conventional electrical circuitry.

Referring now to FIGS. 1 and 2, a block diagram of amodulator/controller 108 in accordance with the present invention isshown. The modulator/controller 108 includes a summing circuit 200, again circuit 204, a modulating circuit 208 and various connections. Theconnections include a first connection to receive a first voltage(output voltage (y)) 110 from the converter 104, a second connection toreceive a second voltage (reference voltage (y₀)) 112 from a referencevoltage source (not shown), a third connection to receive a firstcurrent (inductor current (x₁)) 114 from the converter 104, a fourthconnection to receive an input voltage (u₀) 116 from the voltage source102 (i.e., the input voltage to the converter 104), and a fifthconnection to output a control signal (d) 118 to the converter 104. Thesumming circuit 200 is connected to the first connection and the secondconnection to create a third voltage (Δy) 202 representing a differencebetween the first voltage (y) 110 and the second voltage (y₀) 112. Thegain circuit 204 is connected to the summing circuit 200 to adjust thethird voltage (Δy) 202 by a proportional gain (k), or by any suitablecontroller, such as proportional (P), integral (I) or derivative (D) (orany combination of these three) controller. The modulating circuit 208is connected to the gain circuit 204, the second connection, the thirdconnection, the fourth connection and the fifth connection. Themodulation circuit 208 creates a control signal (d) 118 that providesleading-edge modulation with input-output linearization based on thefirst voltage (y) 110, the second voltage (y₀) 112, the adjusted thirdvoltage (kΔy) 206, the first current (x₁) 114 and the fourth voltage(u₀) 116. Whenever the converter 104 is a boost converter, the controlsignal (d) 118 has a duty cycle defined by

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$Whenever the converter 104 is a buck-boost converter, the control signal(d) 118 has a duty cycle defined by

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$

The present invention also provides an apparatus having one or moreelectrical circuits that provide a control signal 118 to a boostconverter such that a duty cycle of the control signal is defined as

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$Similarly, the present invention provides an apparatus having one ormore electrical circuits that provide a control signal 118 to abuck-boost converter such that a duty cycle of the control signal isdefined as

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$In either case, the apparatus may include a summing circuit, a gaincircuit, a modulating circuit and various connections. The connectionsinclude a first connection to receive a first voltage (y) 110 from theconverter 104, a second connection to receive a second voltage (y₀) 112from a reference voltage source, a third connection to receive a firstcurrent (x₁) 114 from the converter 104, a fourth connection to receivea fourth voltage (u₀) 116 from the voltage source 102 (i.e., the inputvoltage to the converter 104), and a fifth connection to output acontrol signal (d) 118 to the converter 104. The summing circuit 200 isconnected to the first connection and the second connection to create athird voltage (Δy) 202 representing a difference between the firstvoltage (y) 110 and the second voltage (y₀) 112. The gain circuit 204 isconnected to the summing circuit 200 to adjust the third voltage (Δy)202 by a proportional gain (k), or by any suitable controller, such asproportional (P), integral (I) or derivative (D) (or any combination ofthese three) controller. The modulating circuit 208 is connected to thegain circuit 204, the second connection, the third connection, thefourth connection and the fifth connection. The modulation circuit 208creates a control signal (d) that provides leading-edge modulation withinput-output linearization based on the first voltage (y) 110 from theoutput of the converter 104, the second voltage (y₀) 112 from thereference voltage source, the adjusted third voltage (kΔy) 206 from thegain circuit 204, the first current (x₁) 114 from the inductor withinthe converter 104 and the fourth voltage (u₀) 116 from the voltagesource 102 (i.e., the input voltage to the converter 104).

The present invention can be sold as a kit for engineers to design andimplement a PWM modulated converter (boost or buck-boost). The kit mayinclude a digital signal processor, or FPGA, and a computer programembodied on a computer readable medium for programming the digitalsignal processor, or FPGA, to control the PWM modulated converter. Thecomputer program may also include one or more design tools. The digitalsignal processor, or FPGA, includes a summing circuit 200, a gaincircuit 204, a modulating circuit 208 and various connections. Theconnections include a first connection to receive a first voltage 110, asecond connection to receive a second voltage 112, a third connection toreceive a first current 114, a fourth connection to receive an inputvoltage 116, and a fifth connection to output a control signal 118. Thesumming circuit 200 is connected to the first connection and the secondconnection to create a third voltage (Δy) 202 representing a differencebetween the first voltage and the second voltage. The gain circuit 204is connected to the summing circuit 200 to adjust the third voltage (Δy)202 by a proportional gain (k) or by any suitable controller, such asproportional (P), integral (I) or derivative (D) (or any combination ofthese three) controller. The modulating circuit 208 is connected to thegain circuit 204, the second connection, the third connection, thefourth connection and the fifth connection. The modulation circuit 208creates a control signal 118 that provides leading-edge modulation withinput-output linearization based on the first voltage (y) 110, thesecond voltage (y₀) 112, the adjusted third voltage (kΔy) 206, the firstcurrent (x₁) 114 and the input voltage (u₀) 116. Whenever the converter104 is a boost converter, the control signal (d) 118 has a duty cycledefined by

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$Whenever the converter 104 is a buck-boost converter, the control signal(d) 118 has a duty cycle defined by

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$As implemented in the system of FIG. 1, the first voltage 110 is anoutput voltage from the converter 104, the second voltage 112 is areference voltage, the first current 114 is an inductor current from theconverter 104 and the fourth voltage 116 is the voltage provided by thevoltage source 102 as the input voltage of the converter 104.

Now referring to FIG. 3A, a flow chart 300 of a control method for aboost converter in accordance with the present invention is shown. Theboost converter is controlled by receiving a first voltage (y) from anoutput of a boost converter, a second voltage (y₀) from a referencevoltage source, a first current (x₁) from an inductor within the boostconverter and a fourth voltage (u₀) from the input of the converter at aPWM modulator/controller in block 302. A third voltage (Δy) is createdrepresenting a difference between the first voltage (y) and the secondvoltage (y₀) in block 304. The third voltage (y₀) is adjusted by aproportional gain (k) or any suitable type of controller, such asproportional (P), integral (I) or derivative (D) (or any combination ofthese three) by replacing k(y₀−y) in the equation defining the dutycycle d with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller in block 306. If k_(i)and k_(d) are both zero, then the controller reduces to a proportionalcontroller. If only k_(d) is zero, then the controller reduces to aproportional-integral (PI) controller. The control signal (d) is createdin block 308 that provides leading-edge modulation with input-outputlinearization based on the first voltage (y), the second voltage (y₀),the adjusted third voltage (kΔy), the first current (x₁) and the fourthvoltage (u₀), wherein the control signal (d) has a duty cycle defined by

$d = {\frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}.}$The boost converter is then controlled using the control signal (d)created by the PWM modulator/controller in block 310. In an optionalembodiment, the boost converter is controlled in block 312 using aproportional controller, or any suitable type of controller, such asproportional (P), integral (I) or derivative (D) (or any combination ofthese three) by replacing k(y₀−y) in the equation defining the dutycycle d with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller. If k_(i) and k_(d) areboth zero, then the controller reduces to a proportional controller. Ifonly k_(d) is zero, then the controller reduces to aproportional-integral (PI) controller. Note that the control signal canbe created using a first order system, or can be independent of astabilizing gain, a desired output voltage or a desired outputtrajectory. Likewise, the present invention may include a computerprogram embodied within a digital signal processor, or FPGA, wherein thesteps are implemented as one or more code segments.

Now referring to FIG. 3B, a flow chart 350 of a control method for abuck-boost converter in accordance with the present invention is shown.The buck-boost converter is controlled by receiving a first voltage (y)from an output of a buck-boost converter, a second voltage (y₀) from areference voltage source, a first current (x₁) from an inductor withinthe buck-boost converter and a fourth voltage (u₀) from voltage sourceproviding an input to the buck-boost converter at a PWMmodulator/controller in block 352. A third voltage (Δy) is createdrepresenting a difference between the first voltage (y) and the secondvoltage (y₀) in block 304. The third voltage (y₀) is adjusted by aproportional gain (k), or any suitable type of controller, such as aproportional (P), integral (I) or derivative (D) (or any combination ofthese three) controller, by replacing k(y₀−y) in the equation definingthe duty cycle d with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller in block 306. If k_(i)and k_(d) are both zero, then the controller reduces to a proportionalcontroller. If only k_(d) is zero, then the controller reduces to aproportional-integral (PI) controller. The control signal (d) is createdin block 354 that provides leading-edge modulation with input-outputlinearization based on the first voltage (y), the second voltage (y₀),the adjusted third voltage (kΔy), the first current (x₁) and the fourthvoltage (u₀), wherein the control signal (d) has a duty cycle defined by

$d = {\frac{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}.}$The buck-boost converter is then controlled using the control signal (d)created by the PWM modulator/controller in block 356. In an optionalembodiment, the buck-boost converter is controlled in block 358 using aproportional controller, or any suitable type of controller, such as aproportional (P), integral (I) or derivative (D) (or any combination ofthese three) controller, by replacing k(y₀−y) in the equation definingthe duty cycle d with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller. If k_(i) and k_(d) areboth zero, then the controller reduces to a proportional controller. Ifonly k_(d) is zero, then the controller reduces to aproportional-integral (PI) controller. Note that the control signal canbe created using a first order system, or can be independent of astabilizing gain, a desired output voltage or a desired outputtrajectory. Likewise, the present invention may include a computerprogram embodied within a digital signal processor, or FPGA, wherein thesteps are implemented as one or more code segments.

A more detailed description of the models used in the present inventionwill now be described. State space averaging allows the adding togetherof the contributions for each linear circuit during its respective timeinterval. This is done by using the duty ratio as a weighting factor oneach interval. As shown below, this weighting process leads to a singleset of equations for the states and the output. But first, the systemwill be described by its state space equations.

Assume that a linear system (A, b) is described by{dot over (x)}(t)=Ax(t)+bu(t)  (1)where Aε

^(n×n) is an ‘n×n’ matrix, and bε

^(n) is an ‘n×1’ column vector.

As previously mentioned, the duty ratio d is the ratio indicating thetime in which a chosen switch is in the “on” position while the otherswitch is in the “off” position. Ts is the switching period. The “on”time is then denoted as dTs. The general state equations for any type ofconverter consisting of two linear switched networks are as follows:

For 0≦t≦dTs,{dot over (x)}(t)=A _(α) x(t)+b _(α) u(t)  (2a)dTs≦t≦Ts,{dot over (x)}(t)=A _(β) x(t)+b _(β) u(t)  (2b)

The equations in (2a) can be combined with the equations in (2b) usingthe duty ratio, d, as a weighting factor. Thus,{dot over (x)}(t)=(dA _(α) +d′A _(β))x(t)+(db _(α) +d′b _(β))u(t)  (3)which can be written in the form of equation (1) as{dot over (x)}(t)=Ax(t)+bu(t)  (4)withA=dA _(α) +d′A _(β)andb=db _(α) +d′b _(β)where d′=1−d.

The buck cell is linear after state-space averaging and is therefore theeasiest topology to control. On the other hand, the boost and buck-boostcells are nonlinear and have non-minimum phase characteristics. Thesenonlinear cells will be described.

Beginning with a vector field f(x) and a scalar function h(x), the Liederivative of h with respect to f is denoted by L_(f)h. The derivativeis a scalar function and can be understood as the directional derivativeof h in the direction of the vector field f.

Definition: For a smooth scalar function b:

^(n)→

and a smooth vector field f:

^(n)→

^(n), the Lie derivative of h with respect to f isL _(f) b=∇bf.  (5)orL _(f) b=<∇b,f>  (6)where ∇ represents the gradient and bold type represents a vector field,∇bf is matrix multiplication, and <∇b,f> is standard dot product on

^(n).

Lie derivatives of any order can be defined asL _(f) ⁰ b=b  (7)L _(f) ^(i) b=∇(L _(f) ^(i−1) b)f=L _(f) L _(f) ^(i−1) b.  (8)Also if g is another smooth vector field g:

^(n)→

^(n), thenL _(g) L _(f) b=∇(L _(f) b)g.  (9)

Now, add an output y to the nonlinear system {dot over (x)}=f(x)+g(x)u,where f(x) and g(x) are C^(∞) vector fields on R^(n). Unlike input-statelinearization where a transformation is first found to generate a newstate vector and a new control input, here the output y is repeatedlydifferentiated until the input u appears, thereby showing a relationshipbetween y and u.

For the nonlinear system{dot over (x)}=f(x)+g(x)uy=b(x)  (10)and a point x₀, we differentiate y once to get{dot over (y)}=∇b{dot over (x)}=∇bf(x)+∇bg(x)u=L _(f) b(x)+L _(g) b(x)u.This is differentiated repeatedly until the coefficient of u isnon-zero. This procedure continues until for some integer r≦nL _(g) L _(f) ^(i) b(x)=0 for all x near x ₀ and 0≦i≦r−2andL _(g) L _(f) ^(r−1) b(x ₀)≠0Then

$\begin{matrix}{u = {- \frac{\left( {{L_{f}^{r}{b(x)}} + v} \right)}{L_{g}L_{f}^{r - 1}{b(x)}}}} & (11)\end{matrix}$and, for v=0, results in a multiple integrator system with transferfunction

$\begin{matrix}{{H(s)} = \frac{1}{s^{r}}} & (12)\end{matrix}$State feedback can be added for pole placement withv=c ₀ b(x)+c ₁ L _(f) b(x)+c ₂ L _(f) ² b(x)+ . . . +c _(r−1) L _(f)^(r−1) b(x),where c₀, c₁, . . . , c_(r−1), are constants to be chosen, and theinteger r is the relative degree of the system (2-10). It is the numberof differentiations required before u appears.

The first r new coordinates are found as above by differentiating theoutput h(x)

$\begin{matrix}{{{\overset{.}{z}}_{1} = {{L_{f}{b\left( {x(t)} \right)}} = z_{2}}}{{\overset{.}{z}}_{2} = {{L_{f}^{2}{b\left( {x(t)} \right)}} = z_{3}}}\ldots{{\overset{.}{z}}_{r - 1} = {{L_{f}^{r - 1}{b\left( {x(t)} \right)}} = z_{r}}}{{\overset{.}{z}}_{r} = {{L_{f}^{r}{b\left( {x(t)} \right)}} + {L_{g}L_{f}^{r - 1}{b\left( {x(t)} \right)}{u(t)}}}}} & (13)\end{matrix}$Since x(t)=Φ⁻¹(z(t)), leta(z)=L _(g) L _(f) ^(r−1) b(Φ⁻¹(z))b(z)=L _(f) ^(r) b(Φ⁻¹(z))which is recognized from (11) that a(z) is the denominator term and b(z)is the numerator. Nowż _(r) =b(z(t))+a(z(t))u(t)where a(z(t)) is nonzero for all z in a neighborhood of z⁰.

To find the remaining n−r coordinates, let

$\xi = \begin{bmatrix}z_{1} \\\vdots \\z_{r}\end{bmatrix}$ and $\eta = {\begin{bmatrix}z_{r + 1} \\\vdots \\z_{n}\end{bmatrix}.}$Here z_(r+1), . . . , z_(n) are added to z₁, . . . , z_(r) to provide alegitimate coordinate system. With this notation we can write the newcoordinates in normal form as

$\begin{matrix}{{{\overset{.}{z}}_{1} = z_{2}}{{\overset{.}{z}}_{2} = z_{3}}\ldots{{\overset{.}{z}}_{r - 1} = z_{r}}{{\overset{.}{z}}_{r} = {{b\left( {\xi,\eta} \right)} + {{a\left( {\xi,\eta} \right)}u}}}{\overset{.}{\eta} = {{q\left( {\xi,\eta} \right)} + {{p\left( {\xi,\eta} \right)}u}}}{y = z_{1}}} & (14)\end{matrix}$

The equation for {dot over (η)} represents the n−r equations for whichno special form exists. The general equation, however, if the followingcondition holdsL _(g)Φ_(i)(x)=0is reduced to{dot over (η)}q(ξ,η)and the input u does not appear.

In general, the new nonlinear system is described by{dot over (ξ)}=Aξ+Bv{dot over (η)}=q(ξ,η)+p(ξ,η)uy=Cξ  (15a,b,c)with the matrices A, B, and C in normal form, andv=b(ξ,η)+a(ξ,η)u

If r=n, input-output linearization leads to input-state linearization.If r<n, then there are n−r equations describing the internal dynamics ofthe system. The zero dynamics, obtained by setting ξ=0 in equation (15b)and solving for η, are very important in determining the possiblestabilization of the system (10). If these zero dynamics are non-minimumphase then the input-output linearization in (11) cannot be used. If,however, the zero dynamics are minimum phase it means that poleplacement can be done on the linear part of (15a) using (11) and thesystem will be stable.

In the sequel, the bold letter used to indicate vector fields will onlybe used when the context is ambiguous as to what is meant. Otherwise,non-bold letters will be used. For a boost converter the driving voltageu(t), the current x₁ through the inductor, and the voltage x₂ across thecapacitor are restricted to be positive, nonnegative, and positive,respectively. Only the continuous conduction mode (CCM) is considered.The duty ratio d is taken to be the control input and is constrained by0≦d≦1. The Ćuk-Middlebrook averaged nonlinear state equations are usedto find a feedback transformation that maps these state equations to acontrollable linear system. This transformation is one-to-one with therestrictions on u(t), x₁ and x₂ just mentioned and with additionalrestrictions involving {dot over (u)}. These additional restrictions arenot needed if u(t) is a constant, as in DC-DC conversion. It isinteresting to note that restrictions on u are unnecessary for the boostconverter even if u(t) changes with time. The nonlinear system is saidto be feedback linearizable or feedback linearized. Through the feedbacktransformation, the same second order linear system for every operatingpoint can be seen.

The new switching model of the present invention will now be describedin more detail. The physical component parasitics R_(s), the DC seriesresistance of the filter inductor L, and R_(c), the equivalent seriesresistance of the filter capacitor C, now need to be included sinceR_(c) especially plays a central role in the analysis to follow.

The system in accordance with the present invention is of the form{dot over (x)}=f(x)+g(x)dy=b(x)  (16)With this in mind, the state equations are derived to include parasiticsR_(s) and R_(c).

There are four basic cells for fixed frequency PWM converters. They arethe buck, boost, buck-boost, and boost-buck (Ćuk) topologies. Manyderivations extend the basic cells in applications where isolation canbe added between input and output via transformers, however, theoperation can be understood through the basic cell. Each cell containstwo switches. Proper operation of the switches results in atwo-switch-state topology. In this regime, there is a controlling switchand a passive switch that are either on or off resulting in two “on”states. In contrast, a three state converter would consist of threeswitches, two controlling switches and one passive switch, resulting inthree “on” states.

The control philosophy used to control the switching sequence ispulse-width-modulation (PWM). A control voltage v_(c) is compared with aramp signal (“sawtooth”), v_(m), and the output pulse width is theresult of v_(c)>v_(m). This is shown in FIG. 4A. A new cycle isinitiated on the negative slope of the ramp. The pulse ends whenv_(c)<v_(m) which causes modulation to occur on the trailing edge. Thisgives it the name “trailing-edge modulation.”

The difference between leading-edge modulation (LEM) and theconventionally used trailing-edge modulation (TEM) is that in TEM (FIG.4A) the pulse-width is determined by the instantaneous control voltagev_(c) prior to switch turn-off, whereas in LEM (FIG. 4B) the pulse-widthis determined by v_(c) prior to switch turn-on. The reason that samplingis “just prior” to switch commutation is that the intersection of v_(c)and v_(m) determines the new state of the switch. Notice that in FIG. 4Bthe sawtooth ramp v_(m) has a negative slope.

Now referring to FIG. 5, a circuit diagram 500 of a boost converter anda modulator/controller 502 in accordance with the present invention isshown. The specifics of the boost converter are well known. In thiscase, S2 is implemented with a diode and S1 is implemented with anN-channel MOSFET. FIGS. 6A and 6B are linear circuit diagrams 600 and650 of the boost converter in FIG. 5 during time DTs and D′Ts,respectively. The converter 500 operates as follows: u₀ provides powerto the circuit during S1 conduction time (FIG. 6A) storing energy ininductor L. During this time S2 is biased off. When S1 turns off, theenergy in L causes the voltage across L to reverse polarity. Since oneend is connected to the input source, u₀, it remains clamped while theother end forward biases diode S2 and clamps to the output. Currentcontinues to flow through L during this time (FIG. 6B). When S1 turnsback on, the cycle repeats. FIG. 7 illustrates the typical waveforms forthe boost converter for the two switched intervals DTs and D′Ts.

The DC transfer function needs to be determined in order to know how theoutput, y, across the load R is related to the input u₀ at zerofrequency. In steady state, the volt-second integral across L is equalto zero. Thus,∫₀ ^(T) ^(s) v _(L) dt=0  (17)where Ts is the switching period.Therefore, the volt-seconds during the on-time must equal thevolt-seconds during the off-time. Using this volt-second balanceconstraint one can derive an equation for volt-seconds during theon-time of S1 (DTs) and another equation for volt-seconds during theoff-time of S1 (D′Ts).

The parasitics are eliminated by setting R_(s)=0 and R_(c)=0.

During time DTs:DT _(s) v _(L) =DT _(s) u ₀  (18)During time D′Ts:D′T _(s) v _(L) =D′T _(s) x ₂ −D′T _(s) u ₀  (19)Since by equation (17)DT _(s) v _(L) =D′T _(s) v _(L)the RHS of equation (18) is set equal to the RHS of equation (19)resulting in

$\begin{matrix}{\frac{x_{2}}{u_{0}} = \frac{1}{D^{\prime}}} & (20)\end{matrix}$Equation (20) is the ideal duty ratio equation for the boost cell. IfR_(s) and R_(c) are both non-zero then

$\begin{matrix}{\frac{y}{u_{0} - {x_{1}R_{s}}} = \frac{1}{D^{\prime}}} & (21)\end{matrix}$

The output y is

$\begin{matrix}{y = {{D^{\prime}\frac{{RR}_{c}}{R + R_{c}}x_{1}} + {\frac{R}{R + R_{c}}x_{2}}}} & (22)\end{matrix}$Now the state space averaged equations are derived during dTs:

${\overset{.}{x}}_{1} = {{\frac{1}{L}u} - {\frac{R_{s}}{L}x_{1}}}$${\overset{.}{x}}_{2} = {{- \frac{1}{{C\left( {R + R_{c}} \right)}\;}}x_{2}}$And during (1−d)Ts:

${\overset{.}{x}}_{1} = {\frac{1}{L}\left\lbrack {{{- \left( {R_{s} + \frac{{RR}_{c}}{R + R_{c}}} \right)}x_{1}} - {\frac{R}{R + R_{c}}x_{2}} + u_{0}} \right\rbrack}$${\overset{.}{x}}_{2} = {{\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} - {\frac{1}{C\left( {R + R_{c}} \right)}x_{2}}}$Combining, the averaged equations are:

$\begin{matrix}{{{{\overset{.}{x}}_{1} = {{\frac{1}{L}u_{0}} - {\frac{R_{s}}{L}x_{1}} - {\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}} - {\frac{R}{L\left( {R + R_{c}} \right)}{x_{2}\left( {1 - d} \right)}}}}\mspace{79mu}{{\overset{.}{x}}_{2} = {{\frac{R}{C\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}} - {\frac{1}{C\left( {R + R_{c}} \right)}x_{2}}}}}\mspace{79mu}{y = {{\frac{R}{\left( {R + R_{c}} \right)}x_{2}} + {\frac{{RR}_{c}}{\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}}}}} & \left( {23a,b,c} \right)\end{matrix}$where R_(s) is the dc resistance of L and R_(c) is the equivalent seriesresistance of C.

In standard form:

$\begin{matrix}{{{\overset{.}{x}}_{1} = {\frac{u_{0}}{L} - {\frac{R}{L\left( {R + R_{c}} \right)}x_{2}} - {\left( {\frac{R_{s}}{L} + \frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}} \right)x_{1}} + {\left( {{\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}x_{1}} + {\frac{R}{L\left( {R + R_{c}} \right)}x_{2}}} \right)d}}}\mspace{79mu}{{\overset{.}{x}}_{2} = {{{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}} + {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} - {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}d}}}\mspace{79mu}{y = {{\frac{{RR}_{c}}{\left( {R + R_{c}} \right)}x_{1}} + {\frac{R}{\left( {R + R_{c}} \right)}x_{2}}}}} & \left( {24a,b,c} \right)\end{matrix}$

Here it is assumed that leading-edge modulation is used so that samplingof the output y only takes place during the interval (1−d)T_(s).Therefore, the weighting factor (1−d) in equation (23c) for y has beenremoved because when the sample is taken the data represents both termsas shown in equation (24c). In the present analysis the effects ofsampling (complex positive zero pair at one-half the sampling frequency)have been ignored.

The input-output linearization for the boost converter will now bediscussed. The output, y, only needs to be differentiated once beforethe control d appears.

$\begin{matrix}{\mspace{79mu}{y = {\frac{R}{R + R_{c}}\left( {x_{2} + {R_{c}x_{1}}} \right)}}} & (25) \\{\mspace{79mu}{\overset{.}{y} = {\frac{R}{R + R_{c}}\left( {{\overset{.}{x}}_{2} + {R_{c}{\overset{.}{x}}_{1}}} \right)}}} & \left( {26a} \right) \\{\overset{.}{y} = {\left( \frac{R}{R + R_{c}} \right)\left( {{\frac{1}{C\left( {R + R_{c}} \right)}\left( {{- x_{2}} + {Rx}_{1} - {{Rx}_{1}d}} \right)} + {\frac{R_{c}}{L}\left( {u_{0} - {\frac{R}{R + R_{c}}x_{2}} - {\left( {R_{s} + \frac{{RR}_{c}}{R + R_{c}}} \right)x_{1}} + {\left( {{\frac{{RR}_{c}}{R + R_{c}}x_{1}} + {\frac{R}{R + R_{c}}x_{2}}} \right)d}} \right)}} \right)}} & \left( {26b} \right)\end{matrix}$

Substituting for x₂ from (25), setting {dot over (y)} equal to k(y₀−y),k>0, and solving for d we get,

$\begin{matrix}{{d = \frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} -} \\{{{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}\end{matrix}}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}}{where}{{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)} > 0.}} & (27)\end{matrix}$Here y₀ is the desired output corresponding to x₁₀ and x₂₀ throughequation (25). The notation has changed and k=c₀ in equation (11), andthe control input is now d instead of u. Here (x₁₀, x₂₀) is anequilibrium point of the boost converter. The proportional term k(y₀−y)can be replaced by any suitable controller, such as a proportional (P),integral (I) or derivative (D) (or any combination of these three) byreplacing k(y₀−y) in the equation defining the duty cycle d with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller. If k_(i) and k_(d) areboth zero, then the controller reduces to a proportional controller. Ifonly k_(d) is zero, then the controller reduces to aproportional-integral (PI) controller.

The control is part of the transformation as shown in equation (27)where it is seen that k(y−y₀) is in the numerator, and k is theproportional gain. The proportional term k(y₀−y) can be replaced by anysuitable controller, such as a proportional (P), integral (I) orderivative (D) (or any combination of these three) by replacing k(y₀−y)in the equation defining the duty cycle d with

$\left( {k_{p} + \frac{k_{i}}{s} + {k_{d}s}} \right)\left( {y_{0} - y} \right)$where k_(p), k_(i), and k_(d) are the gains of the proportional,integral, and derivative terms of the controller. If k_(i) and k_(d) areboth zero, then the controller reduces to a proportional controller. Ifonly k_(d) is zero, then the controller reduces to aproportional-integral (PI) controller. The control implementation isshown in FIG. 2.

Local linearization of the boost converter will now be discussed toobtain a transfer function. A Taylor Series linearization is used on thenonlinear system (24abc) to linearize about an operating point, x₁₀,x₂₀, D and obtain the transfer function. We let{circumflex over (x)} ₁ =x ₁ −x ₁₀ , {circumflex over (x)} ₂ =x ₂ −x ₂₀, ŷ=y−y ₀ , {circumflex over (d)}=d−D.which gives

${\overset{.}{\hat{x}}}_{1} = {\frac{1}{L}\left\lbrack {{{- \frac{\left( {1 - D} \right)R}{R + R_{c}}}{\hat{x}}_{2}} - {\frac{\left( {1 - D} \right){RR}_{c}}{R + R_{c}}{\hat{x}}_{1}} - {R_{s}{\hat{x}}_{1}} + {\left( {{\frac{{RR}_{c}}{R + R_{c}}x_{10}} + {\frac{R}{R + R_{c}}x_{20}}} \right)\hat{d}}} \right\rbrack}$$\mspace{79mu}{{\overset{.}{\hat{x}}}_{2} = {\frac{1}{C\left( {R + R_{c}} \right)}\left\lbrack {{- {\hat{x}}_{2}} + {\left( {1 - D} \right)R{\hat{x}}_{1}} - {{Rx}_{10}\hat{d}}} \right\rbrack}}$In matrix form

$\begin{bmatrix}{\overset{.}{\hat{x}}}_{1} \\{\overset{.}{\hat{x}}}_{2}\end{bmatrix} = {{\begin{bmatrix}{- \left( {\frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)} + \frac{R_{s}}{L}} \right)} & {- \frac{\left( {1 - D} \right)R}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {- \frac{1}{C\left( {R + R_{c}} \right)}}\end{bmatrix}\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{bmatrix}} + {\begin{bmatrix}\frac{{{RR}_{c}x_{10}} + {Rx}_{20}}{L\left( {R + R_{c}} \right)} \\{- \frac{{Rx}_{10}}{C\left( {R + R_{c}} \right)}}\end{bmatrix}{\hat{d}.}}}$Making the following substitutions, which can be derived by letting {dotover (x)}₁=0, {dot over (x)}₂=0, x₁=x₁₀, x₂=x₂₀, R_(c)=0, and R_(s)=0 in(24ab):

$x_{10} = \frac{u_{0}}{\left( {1 - D} \right)^{2}R}$ and$x_{20} = \frac{u_{0}}{\left( {1 - D} \right)}$results in

$\begin{matrix}{\begin{bmatrix}{\overset{.}{\hat{x}}}_{1} \\{\overset{.}{\hat{x}}}_{2}\end{bmatrix} = {{\begin{bmatrix}{- \left( \frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)} \right)} & {- \frac{\left( {1 - D} \right)R}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {- \frac{1}{C\left( {R + R_{c}} \right)}}\end{bmatrix}\;\left\lbrack \begin{matrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{matrix} \right\rbrack} + {\quad\;{\left\lbrack \begin{matrix}{\frac{u_{0}}{{L\left( {R + R_{c}} \right)}\left( {1 - D} \right)}\left( {\frac{R_{c}}{\left( {1 - D} \right)} + R} \right)} \\{- \frac{u_{0}}{{C\left( {R + R_{c}} \right)}\left( {1 - D} \right)^{2}}}\end{matrix} \right\rbrack{\hat{d}.}}}}} & (28) \\{\mspace{79mu}{\hat{y} = {\left\lbrack {\frac{{RR}_{c}}{R + R_{c}}\mspace{14mu}\frac{R}{R + R_{c}}} \right\rbrack\hat{x}}}} & (29)\end{matrix}$

Now a linear system is provided{circumflex over ({dot over (x)})}A{circumflex over (x)}+B{circumflexover (d)}ŷ=C{circumflex over (x)}c  (30)where A is an n×n matrix, B is an n-column vector, and C is an n-rowvector.To find the control-to-output transfer function, solve the matrixequationG(s)=C[sI−A] ⁻¹ B.and obtain

${G(s)} = {{{\frac{1}{\Delta(s)}\left\lbrack {\frac{{RR}_{c}}{R + R_{c}}\mspace{14mu}\frac{R}{R + R_{c}}} \right\rbrack}\begin{bmatrix}{s + \frac{1}{C\left( {R + R_{c}} \right)}} & {- \frac{\left( {1 - D} \right)R}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {s + \left( \frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right.} \right)}\end{bmatrix}}{\quad\left\lbrack \begin{matrix}{\frac{u_{0}}{{L\left( {R + R_{c}} \right)}\left( {1 - D} \right)}\left( {\frac{R_{c}}{\left( {1 - D} \right)} + R} \right)} \\{- \frac{u_{0}}{{C\left( {R + R_{c}} \right)}\left( {1 - D} \right)^{2}}}\end{matrix} \right\rbrack}}$

If we let powers greater than one of R_(c) equal zero, furtherevaluation results in

$\begin{matrix}{{G(s)} = {\frac{1}{\Delta(s)}{\quad\left\lbrack {\frac{u_{0}}{{{LCR}\left( {1 - D} \right)}^{2}}\left\lbrack {{\left( {{{RR}_{c}{C\left( {1 - D} \right)}} - L} \right)s} + {2{R_{c}\left( {1 - D} \right)}} + {R\left( {1 - D} \right)}^{2}} \right\rbrack} \right\rbrack}}} & (31)\end{matrix}$where Δ(s) is the determinant of is [sI−A] which is

$\begin{matrix}{{\Delta(s)} = {s^{2} + {\frac{L + {{RR}_{c}{C\left( {1 - D} \right)}}}{{LC}\left( {R + R_{c}} \right)}s} + \frac{{R\left( {1 - D} \right)}\left( {R_{c} + {R\left( {1 - D} \right)}} \right)}{{{LC}\left( {R + R_{c}} \right)}^{2}}}} & (32)\end{matrix}$Taking the term in (31) associated with s, the zero of the linear systemneeds to be in the left-half plane, so this term needs to remainpositive. Solving for R_(c)C we have

$\begin{matrix}{{R_{c}C} > \frac{L}{R\left( {1 - D} \right)}} & (33)\end{matrix}$Note that the inequality (33) can also be derived from the denominatorof (27) by setting (R+R_(c))=R, i.e., R>>R_(c) and with the followingsubstitution for x₁,

$x_{1} = {\frac{y}{R\left( {1 - D} \right)}.}$

At this point, the transfer function has been shown to be the linearapproximation of the nonlinear system having a left-half plane zerounder constraint (33). The zeros of the transfer function of the linearapproximation of the nonlinear system at x=0 coincide with theeigenvalues of the linear approximation of the zero dynamics of thenonlinear system at η=0. Therefore, the original nonlinear system (24)has asymptotically stable zero dynamics. Furthermore, the followingproposition is associated with the system (14).

Proposition. Suppose the equilibrium n=0 of the zero dynamics of thesystem is locally asymptotically stable and all the roots of thepolynomial p(s) have negative real part. Then the feedback law

$\begin{matrix}{u = {\frac{1}{a\left( {\xi,\eta} \right)}\left( {{- {b\left( {\xi,\eta} \right)}} - {c_{o}{??}_{1}} - {c_{1}{??}_{2}} - \ldots - {c_{r - 1}{??}_{r}}} \right)}} & (34)\end{matrix}$locally asymptotically stabilizes the equilibrium (ξ, η)=(0, 0).

The polynomialp(s)=s ^(r) +c _(r−1) s ^(r−1) + . . . +c ₁ s+c ₀  (35)is the characteristic polynomial of the matrix A associated with theclosed loop system (see equations (14) and (34) and recall that z=ξ){dot over (ξ)}=Aξ+Bv{dot over (η)}=q(ξ,η)  (36)where {dot over (ξ)}=Aξ+Bv are the linear part of the system and {dotover (η)}=q(0,η) are the zero dynamics. The matrix A is given by

$A = {\begin{bmatrix}0 & 1 & 0 & \ldots & 0 \\0 & 0 & 1 & \ldots & 0 \\\vdots & \vdots & \vdots & \vdots & \vdots \\0 & 0 & 0 & \ldots & 1 \\{- c_{o}} & {- c_{1}} & {- c_{2}} & \ldots & {- c_{r - 1}}\end{bmatrix}.}$and the vector B is given byB=[0, . . . ,0,1]^(T).

The feedback law in equation (34) can be expressed in the originalcoordinates as

$\begin{matrix}{u = {\frac{1}{L_{g}L_{f}^{r - 1}{b(x)}}\left( {{{- L_{f}^{r}}{b(x)}} - {c_{o}{b(x)}} - {c_{1}L_{f}{b(x)}} - \ldots - {c_{r - 1}L_{f}^{r - 1}{b(x)}}} \right)}} & (37)\end{matrix}$

As shown in equation (26), the input d appears after only onedifferentiation so the relative degree is one. This means that thepresent invention is a single order linear system containing only oneroot, thus the present invention can be expressed in the new coordinatesas{dot over (ξ)}=−kξ+v{dot over (η)}=q(ξ,η)y=ξ

The polynomial p(s) is simply p(s)=s+k, with k>0, so that thedenominator is now a real pole in the open left half plane.

In accordance with the Proposition, the root of the polynomial p(s) hasa negative real part, and as shown above, the present invention hasasymptotically stable zero dynamics. Therefore, it can be concludedthat, given a control law of the form (37), the original nonlinearsystem (24) is locally asymptotically stable.

The following theorem has been proven.

Theorem 1: For a boost converter with asymptotically stable zerodynamics (using leading-edge modulation),

with constraint

$\begin{matrix}{{R_{c}C} > \frac{L}{R\left( {1 - D} \right)}} & (38)\end{matrix}$and control law

$\begin{matrix}{{d = {\frac{1}{L_{g}L_{f}^{r - 1}y}\left( {{{- L_{f}^{r}}y} - {k\left( {y - y_{o}} \right)}} \right)}};} & (39)\end{matrix}$the nonlinear system{dot over (ξ)}=−kξ+v{dot over (η)}q(ξ,η),y=ξ  (40)with v=0, is asymptotically stable at each equilibrium point (thecharacteristic polynomial p(s) has a root with negative real part) whichmeans that the original nonlinear system{dot over (x)}=f(x)+g(x)dy=b(x)  (41)is locally asymptotically stable at each equilibrium point (x₁₀, x₂₀) inthe setS={(x ₁ ,x ₂),xε

^(n) :x ₁≧0,x ₂>0}with 0≦d≦<1.

Recall that (x₁₀, x₂₀) corresponds to y₀ through equation (25). Theorem1 indicates local asymptotic stability. In practice, the reference inputy₀ is ramped up in a so-called “soft-start” mode of operation. Thistheorem also guarantees local asymptotic stability at each operatingpoint passed through by the system on its way up to the desiredoperating point.

Now referring to FIG. 8, a circuit diagram 800 of a buck-boost converterand a modulator/controller 802 in accordance with the present inventionare shown. The details of buck-boost converters are well known. In thiscase, S2 is implemented with a diode and S1 is implemented with anN-channel MOSFET. FIGS. 9A and 9B are linear circuit diagrams 900 and950 of a buck-boost converter during time DTs and D′Ts. The operation ofthe converter is as follows: u₀ provides power to the circuit during S1conduction time (FIG. 9A) storing energy in inductor L. During this timeS2 is biased off. When S1 turns off, the energy in L causes the voltageacross L to reverse polarity. Since one end is connected to circuitreturn, it remains clamped while the other end forward biases diode S2and clamps to the output. Current continues to flow through L duringthis time (FIG. 9B). When S1 turns back on, the cycle repeats. It shouldbe noted that the output voltage is inverted, i.e., negative. FIG. 10 isa graph of typical waveforms for the buck-boost converter for the twoswitched intervals DTs and D′Ts. A typical embodiment of the buck-boostconverter where the output voltage is positive is the “flyback”converter where a transformer with phase reversal is used instead of aninductor.

It is again desirable to find the DC transfer function to know how theoutput, y, across the load R is related to the input u₀ at zerofrequency. In steady state, the volt-second integral across L is againequal to zero. Thus,∫₀ ^(T) ^(s) v ₁ ,dt=0  (42)where Ts is the switching period.

Therefore, the volt-seconds during the on-time must equal thevolt-seconds during the off-time. Using this volt-second balanceconstraint one can derive an equation for volt-seconds during theon-time of S1 (DTs) and another equation for volt-seconds during theoff-time of S1 (D′Ts).

The parasitics are eliminated by setting R_(s)=0 and R_(c)=0.

During time DTs:DT _(s) v _(L) =DT _(s) u ₀  (43)During time D′Ts:D′T _(s) v _(L) =D′T _(s) x ₂  (44)

Since by equation (42)DT _(s) v _(L) =D′T _(s) v _(L)The RHS of equation (43) is set equal to the RHS of equation (44) toprovide

$\begin{matrix}{{\frac{x_{2}}{u_{0}} = \frac{D}{D^{\prime}}}{y = {- x_{2}}}} & (45)\end{matrix}$

Equation (45) is the ideal duty ratio equation for the buck-boost cell.If R_(s) and R_(c) are both non-zero then

$\begin{matrix}{{\frac{y}{u_{0}} - \frac{x_{1}R_{s}}{D^{\prime}u_{0}}} = {- \frac{D}{D^{\prime}}}} & (46)\end{matrix}$The output y is

$\begin{matrix}{y = {{{- D^{\prime}}\frac{{RR}_{c}}{R + R_{c}}x_{1}} - {\frac{R}{R + R_{c}}x_{2}}}} & (47)\end{matrix}$Once again it is seen in equation (46) that parasitic R_(s) should beminimized. For example if R_(s)=0 and R_(c)=0, then equation (46)reduces to the ideal equation (45).

Now the state space averaged equations are derived during dTs:

${\overset{.}{x}}_{1} = {{\frac{1}{L}u} - {\frac{R_{s}}{L}x_{1}}}$${\overset{.}{x}}_{2} = {{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}}$And during (1−d)Ts:

${\overset{.}{x}}_{1} = {{{- \frac{R_{s}}{L}}x_{1}} - {\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}x_{1}} - {\frac{R}{L\left( {R + R_{c}} \right)}x_{2}}}$${\overset{.}{x}}_{2} = {{\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} - {\frac{1}{C\left( {R + R_{c}} \right)}x_{2}}}$Combining, the averaged equations are:

$\begin{matrix}{{\overset{.}{x}}_{1} = {{\frac{1}{L}u_{0}d} - {\frac{R_{s}}{L}x_{1}} - {\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}} - {\frac{{Rx}_{2}}{L\left( {R + R_{c}} \right)}\left( {1 - d} \right)}}} & \left( {48a} \right) \\{\mspace{79mu}{{{\overset{.}{x}}_{2} = {{\frac{R}{C\left( {R + R_{c}} \right)}{x_{1}\left( {1 - d} \right)}} - {\frac{1}{C\left( {R + R_{c}} \right)}x_{2}}}}\mspace{79mu}{y = {{{- \frac{{RR}_{c}}{\left( {R + R_{c}} \right)}}{x_{1}\left( {1 - d} \right)}} - {\frac{R}{\left( {R + R_{c}} \right)}x_{2}}}}}} & \left( {49b,c} \right)\end{matrix}$In standard form:

$\begin{matrix}{{{\overset{.}{x}}_{1} = {{{- \frac{R}{L\left( {R + R_{c}} \right)}}x_{2}} - {\left( {\frac{R_{s}}{L} + \frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}} \right)x_{1}} + {\left( {{\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}x_{1}} + {\frac{R}{L\left( {R + R_{c}} \right)}x_{2}} + \frac{u_{0}}{L}} \right)d}}}\mspace{79mu}{{\overset{.}{\; x}}_{2} = {{{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}} + {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} - {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}d}}}\mspace{79mu}{y = {{{- \frac{{RR}_{c}}{\left( {R + R_{c}} \right)}}x_{1}} - {\frac{R}{\left( {R + R_{c}} \right)}x_{2}}}}} & \left( {{50\; a},b,c} \right)\end{matrix}$

Here it is assumed that leading-edge modulation is used so that samplingof the output y only takes place during the interval (1−d)T_(s).Therefore, the weighting factor (1−d) in equation (49c) for y has beenremoved in equation (50c) because when the sample is taken the datarepresents both terms. In the present analysis the effects of sampling(complex positive zero pair at one-half the sampling frequency) havebeen ignored.

The output, y, only needs to be differentiated once before the control dappears. Thus,

$\begin{matrix}{\mspace{79mu}{y = {\frac{R}{R + {Rc}}\left( {{{- R_{c}}x_{1}} - x_{2}} \right)}}} & (51) \\{\mspace{79mu}{\overset{.}{y} = {\frac{R}{R + {Rc}}\left( {{{- R_{c}}{\overset{.}{x}}_{1}} - {\overset{.}{x}}_{2}} \right)}}} & \left( {52a} \right) \\{\overset{.}{y} = {\frac{R}{R + {Rc}}\left( {{- {R_{c}\left( {{{- \frac{R}{L\left( {R + R_{c}} \right)}}x_{2}} - {\left( {\frac{R_{s}}{L} + \frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}} \right)x_{1}} + {\left( {{\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}x_{1}} + {\frac{R}{L\left( {R + R_{c}} \right)}x_{2}} + \frac{u_{0}}{L}} \right)d}} \right)}} - \left( {{{- \frac{1}{C\left( {R + R_{c}} \right)}}x_{2}} + {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}} - {\frac{R}{C\left( {R + R_{c}} \right)}x_{1}d}} \right)} \right)}} & \left( {52b} \right)\end{matrix}$Substituting for x₂ from (51), setting {dot over (y)} equal to k(y₀−y),k>0, and solving for d provides,

$\begin{matrix}{{d = \frac{\begin{matrix}{{\left( {{{RR}_{c}C} + L} \right)y} + {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} +} \\{\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}\end{matrix}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)}}{where}{{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}} - {R_{c}{Cu}_{0}}} \right)} > 0.}} & (53)\end{matrix}$Here y₀ is the desired output corresponding to x₁₀ and x₂₀ throughequation (51). The notation has changed and k=c₀ in equation (11), andthe control input is now d instead of u. Here (x₁₀, x₂₀) is anequilibrium point of our buck-boost converter. Implementation of thecontrol is the same as shown in FIG. 2. The same definitions are used,so the local linearization will be discussed.

To obtain the transfer function, a Taylor Series linearization is againused on the nonlinear system (50) to linearize about an operating point,x₁₀, x₂₀, D to provide{circumflex over (x)} ₁ =x ₁ −x ₁₀ , {circumflex over (x)} ₂ =x ₂ −x ₂₀, ŷ=y−y ₀ , {circumflex over (d)}=d−D.This gives

${\overset{.}{\hat{x}}}_{1} = {{{- \left( {\frac{R_{s}}{L} + {\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}\left( {1 - D} \right)}} \right)}{\hat{x}}_{1}} - {\frac{R}{L\left( {R + R_{c}} \right)}\left( {1 - D} \right){\hat{x}}_{2}} + {\left( {{\frac{{RR}_{c}}{L\left( {R + R_{c}} \right)}x_{10}} + {\frac{R}{L\left( {R + R_{c}} \right)}x_{20}} + \frac{u_{0}}{L}} \right)\hat{d}}}$$\mspace{20mu}{{\overset{.}{\hat{x}}}_{2} = {{\frac{R}{C\left( {R + R_{c}} \right)}\left( {1 - D} \right){\hat{x}}_{1}} - {\frac{1}{C\left( {R + R_{c}} \right)}{\hat{x}}_{2}} - {\frac{R}{C\left( {R + R_{c}} \right)}x_{10}\hat{d}}}}$In matrix form

$\begin{bmatrix}{\overset{.}{\hat{x}}}_{1} \\{\overset{.}{\hat{x}}}_{2}\end{bmatrix} = \begin{matrix}{{\begin{bmatrix}{- \left( {\frac{R_{s}}{L} + \frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)}} \right)} & {- \frac{R\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {- \frac{1}{C\left( {R + R_{c}} \right)}}\end{bmatrix}\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{bmatrix}} +} \\{\begin{bmatrix}{\frac{{RR}_{c}x_{10}}{L\left( {R + R_{c}} \right)} + \frac{{Rx}_{20}}{L\left( {R + R_{c}} \right)} + \frac{u_{0}}{L}} \\{- \frac{{Rx}_{10}}{C\left( {R + R_{c}} \right)}}\end{bmatrix}{\hat{d}.}}\end{matrix}$Making the following substitutions, which can be derived by letting {dotover (x)}₁=0, {dot over (x)}₂=0, x₁=x₁₀, x₂=x₂₀, R_(c)=0, and R_(s)=0 in(50ab):

$\begin{matrix}{{x_{10} = \frac{{Du}_{0}}{\left( {1 - D} \right)^{2}R}}{and}{x_{20} = \frac{{Du}_{0}}{\left( {1 - D} \right)}}{{{to}\mspace{14mu}{{get}\begin{bmatrix}{\overset{.}{\hat{x}}}_{1} \\{\overset{.}{\hat{x}}}_{2}\end{bmatrix}}} = \begin{matrix}{{\begin{bmatrix}{- \left( {\frac{R_{s}}{L} + \frac{{RR}_{c}\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)}} \right)} & {- \frac{R\left( {1 - D} \right)}{L\left( {R + R_{c}} \right)}} \\\frac{R\left( {1 - D} \right)}{C\left( {R + R_{c}} \right)} & {- \frac{1}{C\left( {R + R_{c}} \right)}}\end{bmatrix}\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2}\end{bmatrix}} +} \\{\begin{bmatrix}{{\frac{{Du}_{0}}{\left( {1 - D} \right)L}\left( {\frac{R_{c}}{\left( {R + R_{c}} \right)\left( {1 - D} \right)} + \frac{R}{\left( {R + R_{c}} \right)}} \right)} + \frac{u_{0}}{L}} \\\frac{u_{0}D}{{C\left( {R + R_{c}} \right)}\left( {1 - D} \right)^{2}}\end{bmatrix}\hat{d}}\end{matrix}}} & (54) \\{\hat{y} = {\begin{bmatrix}{- \frac{{RR}_{c}}{R + R_{c}}} & {- \frac{R}{R + R_{c}}}\end{bmatrix}\hat{x}}} & (55)\end{matrix}$

Now the linear system{circumflex over ({dot over (x)})}=A{circumflex over (x)}+B{circumflexover (d)}ŷ=C{circumflex over (x)}  (56)is provided where A is an n×n matrix, B is an n-column vector, and C isan n-row vector.To find the control-to-output transfer function the matrix equation issolvedG(s)=C[sI−A] ⁻¹ B.After some algebra, letting powers greater than one of R_(c) equal zeroprovides

$\begin{matrix}{{G(s)} = {\frac{1}{\Delta(s)}\left\{ {{- \frac{u_{0}}{{{LCR}\left( {1 - D} \right)}^{2}}}\left( {{\left\lbrack {{{RR}_{c}{C\left( {1 - D} \right)}} - {LD}} \right\rbrack s} + {\left( {1 + D^{2}} \right)R_{c}} + {R\left( {1 - D} \right)}^{2}} \right)} \right\}}} & (57)\end{matrix}$where Δ(s) is the determinant of [sI−A] which is

$\begin{matrix}{{\Delta(s)} = {s^{2} + {\left( \frac{L + {{RR}_{c}{C\left( {1 - D} \right)}}}{{LC}\left( {R + R_{c}} \right)} \right)s} + \frac{{R\left( {1 - D} \right)}\left\lbrack {R_{c} + {R\left( {1 - D} \right)}} \right\rbrack}{{{LC}\left( {R + R_{c}} \right)}^{2}}}} & (58)\end{matrix}$

Taking the term in (57) associated with s, the zero of the linearapproximation of the system should be in the left-half plane, so thisterm needs to remain positive. Solving for R_(c)C results in

$\begin{matrix}{{R_{c}C} > {\frac{LD}{R\left( {1 - D} \right)}.}} & (59)\end{matrix}$Note that the inequality (59) can also be derived from the denominatorof (53) by setting (R+R_(c))=R, i.e., R>>R_(c) and with the followingsubstitutions for x₁ and u₀,

$x_{1} = {- \frac{y}{R\left( {1 - D} \right)}}$ and$u_{0} = {{- y}{\frac{\left( {1 - D} \right)}{D}.}}$

At this point, it has been shown that the transfer function of thelinear approximation of the nonlinear system has a left-half plane zerounder constraint (59). As before, it is known that the zeros of thetransfer function of the linear approximation of the nonlinear system atx=0 coincide with the eigenvalues of the linear approximation of thezero dynamics of the nonlinear system at η=0. Therefore, the originalnonlinear system (50) has asymptotically stable zero dynamics.

The Proposition is again used, with p(s) as above in equation (35) andthe closed loop system as in equation (36). As shown in equation (52),the input d appears after only one differentiation so the relativedegree is again one. This means that the present invention is a singleorder linear system containing only one root, thus the present inventioncan be expressed in the new coordinates as{dot over (ξ)}=−kξ+v{dot over (η)}=q(ξ,η)y=ξThe polynomial p(s) is simply p(s)=s+k, with k>0, so that thedenominator is now a real pole in the open left half plane.

In accordance with the Proposition, the root of the polynomial p(s) hasa negative real part, and as shown above, the present invention hasasymptotically stable zero dynamics. Therefore, given a control law ofthe form (37), it can be concluded that the original nonlinear system(50) is locally asymptotically stable.

The following theorem has been proven.

Theorem 2: For a buck-boost converter with asymptotically stable zerodynamics (using leading-edge modulation), with constraint

$\begin{matrix}{{R_{c}C} > \frac{LD}{R\left( {1 - D} \right)}} & (60)\end{matrix}$and control law

$\begin{matrix}{{d = {\frac{1}{L_{g}L_{f}^{r - 1}y}\left( {{{- L_{f}^{r}}y} - {k\left( {y - y_{0}} \right)}} \right)}};} & (61)\end{matrix}$the nonlinear system{dot over (ξ)}=−kξ+v{dot over (η)}q(ξ,η),y=ξ  (62)with v=0, is asymptotically stable at each equilibrium point (thecharacteristic polynomial p(s) has a root with negative real part) whichmeans that the original nonlinear system{dot over (x)}=f(x)+g(x)dy=b(x)  (63)is locally asymptotically stable at each equilibrium point (x₁₀, x₂₀) inthe setS={(x ₁ ,x ₂)xε

^(n) :x ₁≧0,x ₂>0}with 0≦d<1.

Recall that (x₁₀, x₂₀) corresponds to y₀ through equation (51).

Theorem 2 indicates local asymptotic stability. In practice, thereference input y₀ is ramped up in a so-called “soft-start” mode ofoperation. This theorem also guarantees local asymptotic stability ateach operating point passed through by the system on its way up to thedesired operating point.

Although preferred embodiments of the present invention have beendescribed in detail, it will be understood by those skilled in the artthat various modifications can be made therein without departing fromthe spirit and scope of the invention as set forth in the appendedclaims.

The invention claimed is:
 1. A system for converting an input voltage toan output voltage, comprising: a voltage converter circuit comprising aninductor, powered by the input voltage and producing an inductor currentthrough the inductor; a controller connected to the voltage convertercircuit, the output voltage, and the input voltage; a reference voltageconnected to the controller; a control signal generated by thecontroller for the voltage converter circuit, comprising a duty cyclebased on the input voltage, the output voltage, the reference voltage,and the inductor current; whereby the voltage converter circuitgenerates the output voltage based on the duty ratio; wherein the dutycycle is given by${d = \frac{{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{c}R_{s}C}} \right){Rx}_{1}} - {{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}}{R\left( {{R_{c}{Cy}} + {\frac{LR}{R + R_{c}}x_{1}}} \right)}};$wherein C is a capacitance value of a capacitor of the voltage convertercircuit, R_(C) is a series capacitive resistance of the capacitor, R isa load resistance, L is an inductance value of an inductor of thevoltage converter circuit, R_(S) is a series resistance of the inductor,and u₀ is the input voltage; and, wherein y₀ is the reference voltage, yis the output voltage, k is a proportional gain factor, and x₁ is theinductor current.
 2. The system of claim 1, wherein the controller isconfigured to generate the control signal based on input-output feedbacklinearization of a set of state variables with stable zero dynamics. 3.The system of claim 1, wherein the controller further comprises: asumming circuit connected to the output voltage and the referencevoltage; a gain circuit connected to the summing circuit; and, amodulating circuit connected to the gain circuit and the output voltage,the reference voltage, the input voltage, and the inductor current, togenerate the control signal.
 4. The system of claim 3, furthercomprising: a difference voltage generated by the summing circuit; anadjusted voltage generated by the gain circuit from the differencevoltage and the proportional gain factor; and, wherein the differencevoltage is the difference between the reference voltage and the outputvoltage.
 5. The system of claim 4, wherein the gain circuit furthercomprises: a gain controller connected to the summing circuit; whereinthe gain controller is selected from the group consisting of aproportional controller, an integral controller, a derivativecontroller, and a combination controller comprising any combination ofthe proportional controller, the integral controller, and the derivativecontroller; and, wherein the proportional gain factor is ( p + i s + d ⁢s ) ⁢ ( 0 - ) , where k_(p), k_(i) and k_(d) are the gains of theproportional, integral, and derivative terms of the gain controller, sis a complex variable, y₀ is the reference voltage, and y is the outputvoltage.
 6. The system of claim 5, wherein the voltage converter circuitis a boost converter.
 7. The system of claim 6, wherein the boostconverter further comprises: the inductor switchably connected in serieswith the load resistance and the input voltage; and, the capacitorconnected in parallel with the load resistance.
 8. In a systemcomprising a voltage converter circuit comprising an inductor, and acontroller connected to the voltage converter circuit, a method forconverting an input voltage to an output voltage comprising the stepsof: receiving a reference voltage; receiving the input voltage;receiving an inductor current; generating a feedback output voltage;receiving the feedback output voltage; generating a control signal fromthe input voltage, the feedback output voltage, the reference voltage,and the inductor current; applying the control signal to the voltageconverter circuit; and, generating the output voltage based on thecontrol signal; wherein the step of generating a control signalcomprises the steps of calculating a duty cycle and solving${d = \frac{\left( {{\left( {{{RR}_{c}C} + L} \right)y} - {\left( {L - {R_{s}R_{c}C}} \right){Rx}_{1}} - {{RR}_{c}{Cu}_{0}} + {\left( {R + R_{c}} \right){{LCk}\left( {y_{0} - y} \right)}}} \right)}{R\left( {{R_{c}{Cy}} - {\frac{LR}{R + R_{c}}x_{1}}} \right)}},$where C is a capacitance value of a capacitor, R_(C) is a seriescapacitive resistance of the capacitor, R is a load resistance, L is aninductance value of an inductor, R_(S) is a series resistance of theinductor, and u₀ is the input voltage, and where y₀ is the referencevoltage, y is the output voltage, k is a proportional gain factor, andx₁ is the inductor current, for the duty cycle.
 9. The method of claim8, further comprising the steps of: creating a difference voltage fromthe feedback output voltage and the reference voltage; and, adjustingthe difference voltage by the proportional gain factor to create anadjusted voltage.
 10. The method of claim 9, wherein the step ofcalculating a duty cycle further comprises the step of implementinginput-output linearization.
 11. The method of claim 9, wherein the stepof generating a control signal further comprises the step of creatingthe control signal based on the output voltage, the reference voltage,the adjusted voltage, the input voltage, and the duty cycle.
 12. Themethod of claim 9, further comprising the step of providing a boostconverter for the voltage converter circuit, wherein the boost converterfurther comprises: the inductor switchably connected in series with theload resistance and the input voltage; and, the capacitor connected inparallel with the load resistance.
 13. A system for converting an inputvoltage to an output voltage, comprising: a voltage converter circuitcomprising an inductor, powered by the input voltage and producing aninductor current through the inductor; a controller connected to thevoltage converter circuit, the output voltage, and the input voltage; areference voltage connected to the controller; a control signalgenerated by the controller for the voltage converter circuit,comprising a duty cycle based on the input voltage, the output voltage,the reference voltage, and the inductor current; whereby the voltageconverter circuit generates the output voltage based on the duty ratio;wherein the duty cycle is given by d = ( R ⁢ ⁢ R c ⁢ C + L ) ⁢ + ( L - R c ⁢R s ⁢ C ) ⁢ R ⁢ 1 + ( R + R c ) ⁢ L ⁢ ⁢ C ⁢ ⁢ ( 0 - ) R ⁡ ( R c ⁢ C ⁢ + L ⁢ ⁢ R R + Rc ⁢ 1 - R c ⁢ C ⁢ ⁢ u 0 ) ; wherein C is a capacitance value of a capacitorof the voltage converter circuit, R_(C) is a series capacitiveresistance of the capacitor, R is the load resistance, L is aninductance value of an inductor of the voltage converter circuit, R_(S)is a series resistance of the inductor, and u₀ is the input voltage;and, wherein y₀ is the reference voltage, y is the output voltage, k isa proportional gain factor, and x₁ is the inductor current.
 14. Thesystem of claim 13, wherein the controller is configured to generate thecontrol signal based on input-output feedback linearization of a set ofstate variables with stable zero dynamics.
 15. The system of claim 13,wherein the controller further comprises: a summing circuit connected tothe output voltage and the reference voltage; a gain circuit connectedto the summing circuit; and, a modulating circuit connected to the gaincircuit and the output voltage, the reference voltage, the inputvoltage, and the inductor current, to generate the control signal. 16.The system of claim 15, further comprising: a difference voltagegenerated by the summing circuit; an adjusted voltage generated by thegain circuit from the difference voltage and the proportional gainfactor; and, wherein the difference voltage is the difference betweenthe reference voltage and the output voltage.
 17. The system of claim16, wherein the gain circuit further comprises: a gain controllerconnected to the summing circuit; wherein the gain controller isselected from the group consisting of a proportional controller, anintegral controller, a derivative controller, and a combinationcontroller comprising any combination of the proportional controller,the integral controller, and the derivative controller; and, wherein theproportional gain factor is ( p + i s + d ⁢ s ) ⁢ ( 0 - ) , where k_(p),k_(i) and k_(d) are the gains of the proportional, integral, andderivative terms of the gain controller, s is a complex variable, y₀ isthe reference voltage, and y is the output voltage.
 18. The system ofclaim 17, wherein the voltage converter circuit is a buck-boostconverter.
 19. The system of claim 18, wherein the buck-boost converterfurther comprises: the inductor switchably connected in series with aload resistance; the input voltage switchably connected in parallel withthe inductor; and, the capacitor connected in parallel with the loadresistance.
 20. In a system comprising a voltage converter circuitcomprising an inductor, and a controller connected to the voltageconverter circuit, a method for converting an input voltage to an outputvoltage comprising the steps of: receiving a reference voltage;receiving the input voltage; receiving an inductor current; generating afeedback output voltage; receiving the feedback output voltage;generating a control signal from the input voltage, the feedback outputvoltage, the reference voltage, and the inductor current; applying thecontrol signal to the voltage converter circuit; and, generating theoutput voltage based on the control signal; wherein the step ofgenerating a control signal comprises the steps of calculating a dutycycle and solving d = ( R ⁢ ⁢ R c ⁢ C + L ) ⁢ + ( L - R c ⁢ R s ⁢ C ) ⁢ R ⁢ ⁢ 1 +( R + R c ) ⁢ L ⁢ ⁢ C ⁢ ⁢ ( 0 - ) R ⁡ ( R c ⁢ C ⁢ ⁢ + L ⁢ ⁢ R R + R c ⁢ - R c ⁢ C ⁢ ⁢ u0 ) , where C is a capacitance value of a capacitor, R_(C) is a seriescapacitive resistance of the capacitor, R is the load resistance, L isan inductance value of an inductor, R_(S) is a series resistance of theinductor, and u₀ is the input voltage, and where y₀ is the referencevoltage, y is the output voltage, k is a proportional gain factor, andx₁ is the inductor current, for the duty cycle.
 21. The method of claim20, further comprising the steps of: creating a difference voltage fromthe feedback output voltage and the reference voltage; and, adjustingthe difference voltage by the proportional gain factor to create anadjusted voltage.
 22. The method of claim 21, wherein the step ofcalculating a duty cycle further comprises the step of implementinginput-output linearization.
 23. The method of claim 21, wherein the stepof generating a control signal further comprises the step of creatingthe control signal based on the output voltage, the reference voltage,the adjusted voltage, the input voltage, and the duty cycle.
 24. Themethod of claim 21, further comprising the step of providing abuck-boost converter for the voltage converter circuit, wherein thebuck-boost converter further comprises: the inductor switchablyconnected in series with the load resistance; the input voltageswitchably connected in parallel with the inductor; and, the capacitorconnected in parallel with the load resistance.